Maximal and essential ideas of MV-algebras

Abstract

We show that an atom free ideal is densely ordered. It is shown that if $I$ is a maximal ideal of an $MV\mbox{-algebra}\;A,$ then =I^\perp\oplus I^{\perp\perp}$ where $I^\perp=\{x\vert x\le e\}$ and $I^{\perp\perp} =\{x\vert x\le \bar e\}$ for a unique idempotent $e.$ The socle, radical and implicative radical of $A$ are computed in certain cases. It is shown that if $A$ is not atom free but $I$ is a maximal ideal which is atom free, then $I$ is densely ordered, and $I=\la At(A)\ra^\perp=\la a\ra^\perp$ where $At(A)$ is the set of atoms of $A$ and $a\in At(A).$ Then $A=I^\perp\oplus I^{\perp\perp}$ where $I^\perp$ is atomic and $I^{\perp\perp}$ is atom free.